Let $(R,m,k)$ be a commutative Noetherian local ring. Suppose $E$ is the injective envelope of $k$. For any module $M$, denote by $M^*=Hom_R(M,E)$. Denote by $R^～$the $m$-adic completion of $R$.
Let $E_n=\{x\in E\mid m^nx=0\}$, then $E_n\cong (R/m^n)^*$ and $E=\varinjlim E_n$, that is, the union of $E_n$ is $E$.
$Hom_R(E,E)\cong (\varinjlim E_n)^*\cong lim_{\leftarrow}(E_n)^*\cong lim_{\leftarrow}(R/m^n)^{**}\cong lim_{\leftarrow}R/m^n=R^～$. Here, we use $(R/m^n)^{**}\cong R/m^n$ since $R/m^n$ has finite length.
Question 1: Is the above isomorphism of $R$-modules a ring isomorphism? If so, how to prove it?
I have found an interesting thing.
Let $\alpha: R\rightarrow R^～$ be the natural embedding. Let $\beta :R\rightarrow End_R(E)$ be the map that sends $r$ to $f_r$, where $f_r(x)=rx$. Denote by $\delta:End_R(E)\rightarrow R^～$ the above isomorphism. I think that, in general, $\delta\beta $ is not equal to $\alpha$. Since the set of associate primes $ass(E)=\{m\}$ and hence $ass(E)=Supp(E)=V(ann(E))$ so if $dimR>0$, then $ann(E)$ is not equal to 0.
Question 2: When is $\beta$ injective?
Question 3: When does the identity $\delta\beta=\alpha$ hold true?